This paper formalizes the relation between comparative statics and the out-of-equilibrium explanation for how a system evolves after a change in parameters. The paper has two main results. First, an increase in an exogenous parameter sets o# learning dynamics that involve larger values of the endogenous variables. Second, equilibrium selections that are not monotone increasing in the exogenous variables must be predicting unstable equilibria. Moreover, under some conditions monotone comparative statics and stability are equivalent. JEL Classification: C61, C62, C72, C73 Keywords: Monotone comparative statics, supermodularity, strategic complements, learning, correspondence principle. # Discussions with Ilya Segal and Chris Shannon were very important for this work, I am very grateful for all their help. For comments and advice, I also thank Robert Anderson, Juan Dubra, Nestor Gandelman, Ernesto Lopez Cordova, Marcelo Moreira, Charles Pugh, Matthew Rabin, Tarun Sabarwal and Miguel Villa...

Firms and workers may sign complex contracts that govern many aspects of their interactions. I show that when firms regard contracts as substitutes, bargaining over contracts can be understood as bargaining only over wages. Substitutes is the assumption commonly used to guarantee the existence of stable matchings of workers and firms. JEL: C78, D86, J31, J41

This paper proposes a method for testing complementarities between explanatory and dependent variables in a large class of economic models. The proposed test is based on the monotone comparative statics (MCS) property of equilibria. Our main result is that MCS produces testable implications on the (small and large) quantiles of the dependent variable, despite the presence of multiple equilibria. The key features of our approach are: (1) we work with a nonparametric structural model of a continuous dependent variable in which the unobservable is allowed to be correlated with the explanatory variable in a reasonably general way; (2) we do not require the structural function to be known or estimable; (3) we remain fairly agnostic on how an equilibrium is selected. We illustrate the usefulness of our result for policy evaluation within Berry, Levinsohn, and Pakes’s (AER, 1999) model.

We propose a theory of intertemporal choice that is robust to specific assumptions on the discount rate. One class of models requires that one utility stream be chosen over another if and only if its discounted value is higher for all discount factors in a set. Another model focuses on an average discount factor. Yet another model is pessimistic, and evaluates a flow by the lowest available discounted value.

We investigate the testable implications of the theory of stable matchings. We provide a characterization of the matchings that are rationalizable as stable matchings when agents' preferences are unobserved. The characterization is a simple nonparametric test for stability, in the tradition of revealed preference tests. We also characterize the observed stable matchings when monetary transfers are allowed and the stable matchings that are best for one side of the market: extremal stable matchings. We find that the theory of extremal stable matchings is observationally equivalent to requiring that there be a unique stable matching or that the matching be consistent with unrestricted monetary transfers.

We revisit the recent revealed preference analysis of sophisticated quasi-hyperbolic consumers by Blow et al. [(2021), “Non-parametric Analysis of Time-Inconsistent Preferences”, The Review of Economic Studies, 88, 2687–2734] (BBC). We show that BBC’s revealed preference test is too lax. There are non-rationalizable data that would pass their test. A basic problem with their test is that it requires finding a certain endogenous elasticity, without regard to the rationalizing utility. Their approach motivates a more stringent test, also based on first-order conditions, that would connect the endogenous elasticity and utility: We show that this test is also too lax. Aside from testing, we also discuss the possibility of recovering model parameters. We show that, even when discount factors are exactly identified, the approach followed in BBC allows for incorrect parameter values to lie in their identified set.

We develop a behavioral axiomatic characterization of Subjective Expected Utility (SEU) under risk aversion. Given is an individual agent’s behavior in the market: assume a finite collection of asset purchases with corresponding prices. We show that such behavior satisfies a “revealed preference axiom” if and only if there exists a SEU model (a subjective probability over states and a concave utility function over money) that accounts for the given asset purchases.

We define the empirical content of an economic theory as the least restrictive observationally equivalent theory. We show that the empirical content of a theory is captured by a certain kind of axiomatization, with axioms that are universal negations of conjunctions of atomic formulae.

We characterize choice rules for schools that regard students as substitutes while expressing preferences for a diverse student body. The stable (or fair) assignment of students to schools requires the latter to regard the former as substitutes. Such a requirement is in conflict with the reality of schools’ preferences for diversity. We show that the conflict can be useful, in the sense that certain unique rules emerge from imposing both considerations. We also provide welfare comparisons for students when different choice rules are employed.

We study preferences estimated from finite choice experiments and provide sufficient conditions for convergence to a unique underlying “true” preference. Our conditions are weak and, therefore, valid in a wide range of economic environments. We develop applications to expected utility theory, choice over consumption bundles, and menu choice. Our framework unifies the revealed preference tradition with models that allow for errors.